Did I Use the Ratio Test Correctly?

[Jac8897]

Homework Statement

Hi
I want to see if you guys could check my work an tell me if i did it correct and also explain me how to cancel "look at the graph"

Homework Equations

The Attempt at a Solution

 

[NastyAccident]

Yes.

When in doubt, set a term a_n equal to what is being summed. In this case, a_n = 2^n/n!.

Now, get the next term in the series so a_n+1. In this case, a_n+1 = 2^(n+1)/(n+1)!.

Then, plug in a_n and a_n+1 into $$|\frac{a_{n+1}}{a_{n}}|$$

Now, the n! cancels because n! is equivalent to n*(n-1)(n-2)(n-3)(n-4). Whereas (n+1)!, is equivalent to (n+1)(n)(n-1)(n-2)(n-3)(n-4).

So, as you can see, the terms to the right of (n+1) are canceled out.
NastyAccident

 

[Jac8897]

Yes.

When in doubt, set a term a_n equal to what is being summed. In this case, a_n = 2^n/n!.

Now, get the next term in the series so a_n+1. In this case, a_n+1 = 2^(n+1)/(n+1)!.

Then, plug in a_n and a_n+1 into $$|\frac{a_{n+1}}{a_{n}}|$$

Now, the n! cancels because n! is equivalent to n*(n-1)(n-2)(n-3)(n-4). Whereas (n+1)!, is equivalent to (n+1)(n)(n-1)(n-2)(n-3)(n-4).

So, as you can see, the terms to the right of (n+1) are canceled out.



NastyAccident

thanks for the explanation was very helpful